I am trying to estimate the following model: $$y=B_0 + B_1x_1 + B_2x_2 + B_3x_3 + e$$

However, I have an omitted variable bias because $x_2$ and $x_3$ are not observed.

Situation 1

If I have an (exogenous) instrument $z_1$, that is only expected to affect $x_1$:

$$y=B_0 + B_1az_1 + B_2x_2 + B_3x_3 + e$$

Will I then still get an unbiased estimate of $x_1$?

Situation 2

What happens if $z_1$ might be correlated with one of the omitted variables (let's say $x_3$), but this effect is due to an expected correlation between $x_1$ and $x_3$?

As an example, a policy ($z_1$) affects trust in the government ($x_1$), however because trust in the government leads (causal) to less protests ($x_3$), $z_1$ is necessarily correlated with $x_3$.

$$y=B_0 + B_1TrustinGovernment + B_2x_2 + B_3Protests + e$$ $$y=B_0 + B_1aPolicy + B_2x_2 + B_3Protests + e$$

In essence, this perhaps boils down to the question whether there should be no correlation between $z_1$ and the omitted variables, or that there should be no expected causal effect on the omitted variables.

As a last point,

Situation 3

If $z_1$ might AFFECT one of the omitted variables (let's say $x_2$), is there anything left that can be done? Can I for example control for other variables which I expect to be correlated with the omitted variable (proxy like approach)?

  • $\begingroup$ I strongly recommend, as I've said before, that you draw causal diagrams - preferably as complete as you can. The power of causal diagram analysis should not be underestimated. You can read, say, The Book of Why, by Pearl and Mackenzie, to see some of its power. A lot of your questions can be answered straight from the diagram! $\endgroup$ Jun 18 '20 at 14:12
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    $\begingroup$ Or if you find the tone of "The Book of Why" too cocky, read Pearl et al. "Causal Inference in Statistics: A Primer" (2016) instead. I found it to be quite helpful. $\endgroup$ Jun 18 '20 at 15:03
  • $\begingroup$ @RichardHardy Indeed! Me, too. Trying to work my way through "Causality", but it's much stiffer going. $\endgroup$ Jun 18 '20 at 17:05
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    $\begingroup$ @AdrianKeister, it is tough to learn from a textbook on one's own. I managed Pearl et al. (2016) with some help from you and the community, but I am delaying any further studies before I have a real need for causal analysis. (So far I have done it for the sheer fun of it.) $\endgroup$ Jun 18 '20 at 17:49
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    $\begingroup$ What does "...exogenous instrument $z_1$...is only expected to affect $x_1$" mean? Your error term (everything you don't observe) is really $B_2x_2 + B_3x_3 + e$. Exogeneity therefore means $z_1$ is uncorrelated with $B_2x_2 + B_3x_3 + e$. If your instrument is correlated with $x_3$, which is a component of the error term, how do you claim empirically that it is exogenous? $\endgroup$
    – Michael
    Jun 18 '20 at 22:30

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